Coarse-Grain Parallelisation of Multi-Implicit Runge-Kutta Methods

A parallel implementation for a multi-implicit Runge-Kutta method (MIRK) with real eigenvalues is decribed. The parallel method is analysed and the algorithm is devised. For the problem with d domains , the amount of work within the s-stage MIRK method, associated with the solution of system, is proportional to (sd) 3 , in contrast to the simple implicit nite diierence method (IFD) where the amount of work is proportional to d 3. However, it is shown that s-stage MIRK admits much greater time steps for the same order of error. Additionally, the proposed parallelisation transforms the system of the dimension sd to s independent subsystems of dimension d. The amount of work for the sequential solution of such systems is proportional to sd 3. The described parallel algorithm enables the solving of each of the s subsystems on a separate processor; nally, the amount of work is again d 3 , but the proot of a larger time step still remains. To test the theory, a comparative example of the 3-D heat transfer in a human heart with 64 3 domains is shown and numerically calculated by 3-stage MIRK.